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arXiv:2512.24798 · Physical Review A

Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space

J. Dai, A. Molochkov, A. J. Niemi, J. Westerholm

quant-phcond-mat.otherSubmitted Dec 31, 2025• Announced Jan 1, 2026• 0 citations• Revised Jun 29, 2026 (v2)
v1 · Dec 31, 2025v2 · Jun 29, 2026
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Abstract

We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational $E$-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the $E$-manifold. We show that its restricted holonomy group is $\mathrm{SU}(2)$, implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a $π/2$ phase gate and a Hadamard-type gate. For two-qubit operations, we outline how linked holonomic cycles in arrays generate a controlled Chern--Simons phase, enabling an entangling controlled-$X$ (CNOT) gate. We present a Ramsey/echo interferometric protocol that measures the Wilson loop trace of the Wilczek--Zee connection for a control cycle, providing a gauge-invariant signature of the non-Abelian holonomy. As a physically realizable demonstrator, we propose bond-length modulations of a Cs($6s$)--Cs($6s$)--Cs($nd_{3/2}$) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the $E$-manifold.

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•Jan 1, 2012•0 citations
We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational EEE-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the EEE-manifold. We show that its restricted holonomy group is SU(2)\mathrm{SU}(2)SU(2), implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a π/2π/2π/2 phase gate and a Hadamard-type gate. For two-qubit operations, we outline how linked holonomic cycles in arrays generate a controlled Chern--Simons phase, enabling an entangling controlled-XXX (CNOT) gate. We present a Ramsey/echo interferometric protocol that measures the Wilson loop trace of the Wilczek--Zee connection for a control cycle, providing a gauge-invariant signature of the non-Abelian holonomy. As a physically realizable demonstrator, we propose bond-length modulations of a Cs(6s6s6s)--Cs(6s6s6s)--Cs(nd3/2nd_{3/2}nd3/2​)
Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the EEE-manifold.
Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024
This report synthesizes theoretical and experimental work on SU(3) symmetry and ℂP² geometry as control manifolds for qutrit quantum computation. It consolidates explicit constructions of Hamiltonians, Berry/Wilczek–Zee connections, and holonomy operators across platforms including superconducting transmons, photonic orbital-mode qutrits, trapped ions, and ultracold atoms. Special emphasis is placed on geometric robustness, non-Abelian holonomies, and the triadic interference structures unique to SU(3). The review unifies Lie algebra geometry, fiber bundle theory, and quantum engineering into a coherent framework for high-dimensional holonomic control.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert spaceH\mathcal{H}Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspaceHgeo\mathcal{H}_{\text{geo}}Hgeo​of the Hilbert-space that is spanned by states{∣Φk⟩}k\{\vert \Phi_{k}\rangle\}_{k}{∣Φk​⟩}k​that fulfill(Φk∣H^∣Φj⟩=0(\Phi_{k}\vert \hat{H}\vert \Phi_{j}\rangle=0(Φk​∣H^∣Φj​⟩=0, whereH^\hat{H}H^is the system's Hamiltonian. Restricting the propagation toHgeo\mathcal{H}_{\text{geo}}Hgeo​ensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expandingHgeo\mathcal{H}_{\text{geo}}Hgeo​and enabling the synthesis of holonomies from higher symmetry groupsU(N)\mathrm{U}(N)U(N)as larger and more capable computational units [6], [7].
We present a geometric framework for holonomic quantum computing in which quantum gates arise from global properties of control manifolds rather than fine-tuned dynamical evolution. Quantum states are modeled as complex projective fibers over a classical control manifold, and adiabatic loops induce unitary gates through Berry and Wilczek–Zee holonomy. Within this setting, we introduce _Quantum Inner State Manifolds_ (QISMs) as symplectic fiber bundles equipped with a natural unitary connection governed by the Fubini–Study form. Using the Ambrose–Singer theorem, we show that generic QISMs generate holonomy groups dense in U(N)U{(N)}U(N), establishing universality. Fault tolerance emerges from global geometric features, providing a robust geometric foundation for quantum gate design.
In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation. A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions. Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described. A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level Λ configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.